In previous lectures we have studied electromagnetic waves
travelling in vacuum. We now extend this treatment to propagation in
non-conducting materials. In particular we will be interested in the propagation
of waves from one media into a second different media (reflection and refraction
of electromagnetic waves in vacuum
equations in a vacuum lead to wave equations for E and B. The
resultant waves propagate with a velocity c=(e0m0)-1/2.|
|E, B and the propagation direction are mutually perpendicular
|E and B are in phase and have amplitudes related by E=cB.|
in LIH non-conducting media
velocity and refractive index
The treatment in this case parallels that in a vacuum except that
we must replace e0 by e0er and m0 by m0mr. Hence the speed of light in the medium, cm, is given
cm=1/(e0erm0mr)1/2 = c/(ermr)1/2
Where c is the speed of light in a vacuum.
However the only materials that have a mr
which is significantly different from 1 are non-LIH ones (e.g. iron). Hence for
most LIH non-conducting materials cm» c/(er)1/2.
The refractive index n of a given material is defined as the speed
of light in vacuum divided by that in the material. Hence
n = c/cm = (ermr)1/2 » (er)1/2
n2 » er
Because er always has a value greater than one, the speed of light in a
material is always less than in a vacuum.
As both er and n may vary strongly with frequency, discrepancies may arise
when comparing values of er (often the static, DC value is used) with n2 (generally
the value appropriate to optical frequencies).
between E and B
In vacuum E=cB.
In a material this is modified to
H=B/(m0mr)»B/m0 as mr»1
Reflection and refraction at the interface between two different
The aim is to establish the properties of electromagnetic waves
when they encounter a plane interface separating two different non-conducting
materials (generally two different dielectrics). In particular equations will be
derived which give the fractions of the incident wave which are reflected and
transmitted at the interface
description of a plane wave not propagating along one of the principal axes
So far we have only considered waves which propagate along one of
the principal axes. For example a plane wave propagating along the x-axis
is described by an equation of the form E=E0sin(kx-wt),
one propagating along the y-axis by E=E0sin(ky-wt) etc.
In the following we will need to consider plane waves which do
not propagate along one of these principal axes. However we can always
resolve the direction of the wave onto two or more of the principal axes.
For example in the figure below the wave lies in the x/y-plane
and propagates in a direction which makes an angle q to the y-axis. There are hence components cosq along the y-axis and sinq
along the x-axis. The equation for
this plane wave is hence of the form
The wave vector, k, in the above equations is given by 2p/l where l is the wavelength of the wave. In addition if the wave propagates in a material with a velocity cm then
where w is the angular frequency of the wave. Hence
conditions for E, D,
B and H at an interface
In the lectures on dielectrics
and magnetic materials it was
shown that, in the absence of free surface charge and conduction currents, the
following boundary conditions existed for E,
D, B and H
and B: Normal components continuous
and H: Tangential components
These boundary conditions will be used below for electromagnetic waves incident at the boundary between two materials.
and direction of reflected and refracted waves
In the figure below a plane electromagnetic wave travelling in a
material of refractive index n1 is incident on the plane boundary
with a second material of refractive index n2. The incident wave
propagates at an angle qi
to the normal of the boundary and has E-
and H-fields of magnitude Ei
and Hi. The E-field
of the incident wave lies in the plane of incidence (the x/y-plane), this is known as the E parallel configuration.
For this configuration the B-
and H-fields must be normal to the
plane of incidence.
In general there will be, in addition to the incident wave, a
reflected wave at an angle qr and a refracted or transmitted wave at an angle qt.
Each of these waves will have E- and H-fields
as shown with amplitudes related by the expression H=nE/m0c where n is the refractive index of the appropriate material.
The E-fields of the three waves are given by the following expressions
Ei0, Er0 and Et0
are the amplitudes of the E-fields
and k1 and k2 and w1 and w2 are the wavevectors and angular frequencies of the waves in the
The boundary condition for E
requires that the tangential component (the component parallel to the boundary)
be continuous. Hence if we evaluate this component on both sides of the boundary
we must obtain the same result.
The tangential component of each E-field is given by the magnitude of the E-field multiplied by cosq, where q is the appropriate angle.
On the side of the boundary in material 1 the total tangential
component of the E-field is the
difference of the components due to the incident and reflected waves (see above
figure). In material 2 there is only the transmitted wave.
Hence the requirement that the tangential component of the E-field
be continuous can be written
indicates that the preceding expression is evaluated at the boundary y=0.
Substituting in (D) the expressions for Ei, Er
and Et given by (A), (B)
and (C) and evaluated for y=0
This equation must hold at all times and for all values of x.
This is only possible if all the coefficients of x
and all the coefficients of t are
equal. This requires
w1=w2 and k1sinqi=k1sinqr=k2sinqt
frequencies of the waves in the two materials are equal w1=w2
angles of incidence and reflection are equal k1sinqi=k1sinqr
incident and transmitted angles are related by k1sinqi=k2sinqt.
However as k1=n1w1/c and k2=n2w2/c
the expression k1sinqi=k2sinqt.
can be written as n1sinqi=n2sinqt.
This is Snell’s law of refraction.|
The above results would have been obtained if E were polarized normal to the plane of incidence (H polarised parallel to the plane of incidence). These results hence apply to any incident wave.
of the reflected and refracted waves
The above procedure provided information on the frequencies and
directions of the incident, reflected and transmitted waves. The next step is to
derive expressions which give their relative amplitudes.
Returning to equation (E) above, which is valid when E
is parallel to the plane of incidence, the spatial and time dependent
components have been shown to be equal. Hence (E) reduces to
Ei0cosqi - Er0cosqi=Et0cosqt
Where qr has been replaced by qi.
In addition the tangential component of the H-field must be continuous at the boundary between the two
materials. For the present case H is
normal to the incident plane so that the tangential component of H
is simply H.
For the tangential component of H to be continuous we must therefore have
But H=nE/m0c so the above can be written as
Equations (F) and (G) can now be used to eliminate either Er0
through by n2
eliminating Er0. From (G)
r// and t//,
which relate the amplitudes of the reflected and transmitted E-fields
to that of the incident E-field, are
known as the reflection and transmission coefficients for E
parallel to the plane of incidence.
The polarisation of E
can also be aligned normal to the plane of incidence (H is now parallel to the plane of incidence). The boundary
conditions that the tangential components of E and H are continuous
Again eliminating either Er0
or Et0 from these two
equations leads to expressions for the E
perpendicular reflection r^ and transmission t^ coefficients:
The equation (H)-(K) are known as the Fresnel relationships or
The signs of these equations give the relative phases of the waves. If positive there is no phase change, if negative there is a p phase change.
of the Fresnel Equations
The properties can be more easily seen if we consider a special
case where one of the materials is air (n=1) and the other has a refractive
Consider the case where the wave is incident from air. Hence n1=1
and n2=n. The Fresnel equations become
The above figure (plotted for n=2)
shows a number of points:
For normal incidence (qi=qt=0) both r// and r^ and t// and t^ have the same magnitudes
the magnitudes of the reflection coefficients tend to 1 (total reflection) and
the magnitude of the transmission coefficients tend to zero (zero transmission).
For a certain angle qB,
known as the Brewster angle, r// becomes zero whereas r^ remains non-zero. For this angle
light can only be reflected with E perpendicular to the plane of incidence.
If unpolarised light is incident at the Brewster angle then the
reflected light will be polarised.
The Brewster angle is found by setting r//=0.
addition Snell’s law gives us
dividing (L) by (M)
Þ cosqisinqi= cosqtsinqt
sin2qi=sin2qt as 2sinqcosq=sin2q
The equality sin2qi=sin2qt implies that either qi=qt or qi=90-qt. The former can not
be correct as we also must have nsinqt=sinqi and n¹1.
Hence at the Brewster angle we must have qi=90-qt
and Snell’s law becomes
but sin(90-q)=cosq hence
propagating from a material into air
We now have n1=n and n2=1. Fresnel’s
equations have the form
In this case the equations for r// and r^ and for t// and t^ are interchanged compared to those for
when the wave is incident from air.
However because nsinqi=sinqt Þqt>qi.
For qi equal to a certain value, the critical angle qc,
sinqt becomes equal to unity and hence qt=90°.
At this point both reflection coefficients become equal to one and
the waves are totally reflected. As the value of sinqt
can not exceed unity, for qi>qc the reflection coefficients remain equal to unity.
Hence for qi>qc all the incident light is reflected. This is known as ‘total
qi=qc occurs when sinqt=1. Hence nsinqc=1
reflection and transmission coefficients
The reflection and transmission coefficients derived above give the
amplitudes of the electric fields associated with the waves.
However in general it is the power of the wave which is measured
experimentally. From Lecture
19 the power density of an electromagnetic wave is given by the
Poynting vector N=ExH.
This has a magnitude EH = nE2/(cm0),
Hence the energy density µnE2
The coefficients which give the fraction of energy reflected or
transmitted at a boundary between two materials equal the appropriate values of
r2 or t2 with the inclusion of the appropriate value(s) of
If R is the power
reflection coefficient at an interface between a material of refractive index n1
and one of n2 then
where r is the
appropriate reflection coefficient given by the Fresnel relationships.
Similarly if T is the
power transmission coefficient
because energy must always be conserved
What are the reflection and transmission power coefficients for
light incident normally from air into a material of refractive index n?
qi=qt=0° and ½r//½=½r^½=(n-1)/(n+1)
velocity and refractive index|
in a material|
description of a plane wave not propagating along one of the principal axes|
and direction of reflected and refracted waves|
of the reflected and refracted waves: the Fresnel equations|
of the Fresnel equations|
angle and total internal reflection|
|Power reflection and transmission coefficients|
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